Write the reaction and relate pressure to extent of reaction

The reaction is $A(g) \rightarrow B(g) + C(g)$ and initially only $A$ is present.

Let the initial pressure of $A$ be $P_0$.

Suppose at time $t$, a fraction $x$ of $A$ has decomposed.

Then:

• pressure of $A$ at time $t$ = $P_0(1-x)$
• pressure of $B$ at time $t$ = $P_0x$
• pressure of $C$ at time $t$ = $P_0x$

So total pressure at time $t$ is $P_t = P_0(1-x)+P_0x+P_0x = P_0(1+x)$

Find total pressure at completion

At $t=\infty$, reaction is complete, so $x=1$.

Therefore, $P_\infty = P_0(1+1)=2P_0$ Hence, $P_0 = \frac{P_\infty}{2}$

Find partial pressure of unreacted $A$ at time $t$ in terms of $P_t$ and $P_\infty$

We need pressure of reactant $A$ at time $t$ because for a first-order reaction, $k = \frac{1}{t}\ln\frac{[A]_0}{[A]_t}$ and for gases at constant temperature and volume, concentration is proportional to partial pressure.

Now, $P_t = P_0(1+x)$ so $x = \frac{P_t}{P_0}-1$

Pressure of $A$ at time $t$: $P_A = P_0(1-x)$

Substitute $x$: $P_A = P_0\left[1-\left(\frac{P_t}{P_0}-1\right)\right]$ $P_A = P_0\left(2-\frac{P_t}{P_0}\right) = 2P_0-P_t$

Since $2P_0=P_\infty$, $P_A = P_\infty - P_t$

Apply first-order rate law

Initially only $A$ is present, so initial pressure of $A$ is $P_{A,0}=P_0=\frac{P_\infty}{2}$

At time $t$, $P_{A,t}=P_\infty-P_t$

Therefore, $k = \frac{1}{t}\ln\frac{P_{A,0}}{P_{A,t}}$ $k = \frac{1}{t}\ln\frac{P_\infty/2}{P_\infty-P_t}$ $k = \frac{1}{t}\ln\frac{P_\infty}{2(P_\infty-P_t)}$

Match with the options

This is exactly: $\boxed{k=\frac{1}{t}\ln\frac{P_\infty}{2(P_\infty-P_t)}}$

So the correct option is B.